The present invention relates to a linear time-invariant system modeling apparatus of the type that, for example, is used to characterize behavior of the linear time-invariant system in a time-domain. The present invention also relates to a method of generating a passive model of a linear time-invariant system from a non-passive model of the linear time-invariant system, the method being of the type that, for example, is used to characterize behavior of the linear time-invariant system in a time-domain.
In the field of electronic design, engineers, for example microwave engineers, need to analyze, design, and simulate active and passive microwave components and systems. In the face of complex communications signal formats, design verification is challenging. Furthermore, whilst designs can be analyzed early in simulation and verified later once a prototype has returned from fabrication, modern pressures upon vendors to respond rapidly to market requirements means that the designer does not necessarily have the luxury of waiting for fabricated hardware to complete a design. Consequently, designers employ an intermediate level of verification to cut overall design time and reduce risk and cost.
In this respect, the field of electronic design automation strives to provide engineers with tools for modeling and analyzing performance of a design or a part thereof. One known suite of analysis tools for so-called high-frequency design is the Advanced Design System (ADS) available from Agilent Technologies, Inc. The ADS is a Computer Aided Design (CAD) program that supports system and RF design engineers when developing all types of Radio Frequency (RF) designs, ranging from, for example, RF/microwave modules to Monolithic Microwave Integrated Circuits (MMICS) for communications, aerospace and other applications.
The ADS provides a set of simulation technologies ranging from frequency-domain circuit simulation to electromagnetic field simulation. A single, integrated, design environment provides system and circuit simulators, along with schematic capture, layout, and verification capability, thereby eliminating interruptions associated with changing design tools mid-cycle during development. Indeed, verification very early in a prototyping cycle for applications is possible Furthermore, the ADS enables designers to characterize fully and optimize designs. In relation to electromagnetic simulation and/or measurement, most commercial simulation and/or measurement tools perform the simulation and/or measurement in the frequency domain. In this respect, a set of frequency points, for example between 0 Hz and 10 GHz, and associated values, for example amplitudes, are sampled for analysis purposes. However, in order to perform a so-called transient simulation, information is required in the context of the time domain, and significant difficulty is associated with converting frequency domain data to the time domain.
Often, it is necessary to model a so-called Linear Time Invariant (LTI) system. Generally speaking, most passive components either alone or in combination constitute members of a group of LTI systems. A physical example of the LTI system is a microstrip transmission line, a known type of electrical circuit designed to transmit efficiently electromagnetic energy at high frequencies where traditional circuits become inefficient, that needs to be designed and analyzed. The LTI system can be represented in the form of a rational function. In this respect, two examples techniques used to identify the rational function are so-called Vector Fitting (VF), for example as described in “Rational Approximation of Frequency Domain Responses by Vector Fitting” (B. Gustavsen and A. Semlyen, IEEE Transactions on Power Delivery, 14:1052-1061, July 1999), and direct interpolation, for example as described in “Generation of Accurate Rational Models of Lossy Systems using the Cauchy Method” (T. K. Sarkar, A. G. Lamperez and M. S. Palma, Microwave and Wireless Components Letters, 14:490-492, October 2004). In both cases, one obtains a rational function (model) in the Laplace domain of the form:
                                          S                          n              ⁢                                                          ⁢              m                                ⁡                      (            s            )                          =                              d                          n              ⁢                                                          ⁢              m                                +                                    ∑                              k                =                1                            K                        ⁢                                          c                k                                  n                  ⁢                                                                          ⁢                  m                                                            s                -                                  p                  k                                      n                    ⁢                                                                                  ⁢                    m                                                                                                          (        1        )            
In equation (1) above, Snm is an element at row n, column m of a scattering matrix. dnm is a constant contribution, pknm represents the poles of the rational function and cknm are residues of the rational function. All elements of the scattering matrix can be fitted to the same set of poles (pknm=pk) or with a separate set of poles can be used in respect of each element of the scattering matrix.
It is, of course, important that, when performing a simulation, the rational function employed is as close to the actual data initially obtained, either empirically or by simulation, for the LTI system being modeled so that behavior of the LTI system is modeled accurately. In order to obtain physically meaningful simulations in the time domain, it is necessary that the model is passive.
Equation (1) above can be represented in a compact matrix form:S(s)=D+C(sI−A)−1B  (2)
Matrix, D, is a square matrix containing the elements dnm. The matrix, C, contains the residues cnm, and matrix, A, is a complex or real matrix containing the poles pnm. The matrix, B, comprises 0's and 1's, mapping the poles and residues to the correct element locations in the scattering matrix, for example as described in “Equivalent SPICE Circuit With Guaranteed Passivity From Non-Passive Models” (A. Lamecki and M. Mrozowski, IEEE Transactions on Microwave Theory and Techniques, 55:526-532, March 2007). Matrix, I, is the identity matrix of the same size as the matrix, A. It should be appreciated that the inversion of (sI−A) is trivial since matrix, A, is a diagonal matrix.
The set of matrices ABCD is referred to as a “state-space representation” of the rational function expressed in equation (1) above, and describes the broadband behavior of the LTI system in both the frequency- and the time-domain. In this respect, for simulation purposes, it is not sufficient for the model to represent the LTI system over the range of data initially obtained, and it is typically necessary to be able to model the broadband behavior of the LTI system being modeled. In order to “map” the rational function, which is a model of reduced-order, to a (non-linear) circuit simulation scenario, one has to represent the rational function (model) in the state-space form:
                    {                                                                                                  x                    .                                    ⁡                                      (                    t                    )                                                  =                                                      Ax                    ⁡                                          (                      t                      )                                                        +                                      Bu                    ⁡                                          (                      t                      )                                                                                                                                                                y                  ⁡                                      (                    t                    )                                                  =                                                      Cx                    ⁡                                          (                      t                      )                                                        +                                      Du                    ⁡                                          (                      t                      )                                                                                                                              (        3        )            
In equation (3), u(t) is an input signal and y(t) is an output signal, where x(t) is a so-called “internal state” of equation (3), x(t) being the first derivative of x(t) with respect to time.
It is assumed that the LTI system is causal and strictly stable. Consequently, all poles, in particular all eigenvalues of the matrix, A, have negative real parts.
As mentioned above, it is necessary for the rational function described above to be passive in order to perform physically meaningful simulations in the time domain. In this respect, the model of the LTI system is passive if no energy is created within the system represented by the model. Consequently, in order to have a passive state, it is required that, in the frequency domain (i.e. s=jω), all singular values, σ(jω), of the scattering matrix, S(s=jω) are smaller than 1.0, as described in “Passivity Enforcement via Perturbation of Hamiltonian Matrices” (S. Grivet-Talocia, IEEE Transactions on Circuits and Systems, 51:1755-1769, September 2004):
                                                                                          max                  ω                                ⁢                                  σ                  ⁡                                      (                    jω                    )                                                              ≤              1.0                                                          ∀              ω                                                          (        4        )            
Clearly, due to the infinite number of frequencies, one cannot check the above criterion of inequality (4) for all possible frequencies. Consequently, a purely algebraic test is described in “Passivity Enforcement via Perturbation of Hamiltonian Matrices” (S. Grivet-Talocia, IEEE Transactions on Circuits and Systems, 51:1755-1769, September 2004) and “A bisection method for computing the H∞ norm of a transfer matrix and related problems” (V. Balakrishnan, S. Boyd and P. Kabamba, Math. Control Signals Syst., 2:207-219, 1989). Indeed, a Hamiltonian matrix can be constructed having the following form:
                    H        =                  [                                                                      A                  -                                                            BR                                              -                        1                                                              ⁢                                          D                      T                                        ⁢                    C                                                                                                                    -                                          BR                                              -                        1                                                                              ⁢                                      B                    T                                                                                                                                            C                    T                                    ⁢                                      Q                                          -                      1                                                        ⁢                  C                                                                                                  -                                          A                      T                                                        +                                                            C                      T                                        ⁢                                          DR                                              -                        1                                                              ⁢                                          B                      T                                                                                                    ]                                    (        5        )            
In Hamiltonian matrix (5), Q=DTD−I and R=DDT−I. Furthermore, it can be proven that the state-space model is guaranteed to be passive if the Hamiltonian matrix, H, has no purely imaginary eigenvalues:Re{λi}≠0.0 ∀λiελ(H)  (6)
The condition set forth in inequality (6) is a sufficient condition, and it has also been proven that if jω0 is an eigenvalue of the Hamiltonian matrix, H, ω0 is a so-called “cross-over frequency” from a passive frequency interval to a non-passive frequency interval or vice-versa where the model lacks passivity.
If the model is not passive, the challenge is therefore to modify the state-space representation, ABCD, so that all eigenvalues of the Hamiltonian matrix, H, have a non-vanishing real part, whilst preserving the overall behavior of the model pre-modification. In this respect, a number of so-called “passivity enforcement” methods are in existence. For example, “Passivity Enforcement via Perturbation of Hamiltonian Matrices” (S. Grivet-Talocia, IEEE Transactions on Circuits and Systems, 51:1755-1769, September 2004) describes a method of directly modifying the Hamiltonian matrix, H, by evaluating a first-order approximation of perturbation of the eigenvalues by modifying the residues. However, this technique is computationally expensive, because an eigenvector determination of the Hamiltonian matrix, H, is required and the Hamiltonian matrix, H, grows in accordance with 2Np2K, Np being the number of ports of the model and K being the number of poles of the model. In “Equivalent SPICE Circuit With Guaranteed Passivity From Non-Passive Models” (A. Lamecki and M. Mrozowski, IEEE Transactions on Microwave Theory and Techniques, 55:526-532, March 2007), instead of perturbing the residues, a method is described that involves perturbation of the poles. However, this technique also requires the eigenvector calculation of the Hamiltonian matrix, H.
“Computer Code for Passivity Enforcement of Rational Macromodels by Residue Perturbation” (B. Gustavsen” IEEE Transactions on Advanced Packaging, 2:209-215, May 2007), “Fast Passivity Verification and Enforcement via Reciprocal Systems for Interconnects With Large Order Macromodels” (R. Achar, D. Saraswat and M. Nakhla, IEEE Transactions on Very Large Scale Integration (VLSI) systems, 15:48-59, January 2007), and “A Fast Algorithm and Practical Considerations for Passive Macromodeling of Measured/Simulated Data, IEEE Transactions on Advanced Packaging, 27:57-70, February 2004), describe different passivity enforcement techniques, but these techniques are also computational very expensive since a constrained optimization problem must be solved iteratively. Consequently, the high computation cost results in increased demand upon a processing resource supporting simulation software and hence, typically, reduced performance speeds. Also, a number of the passivity enforcement techniques described are not always capable of ensuring passivity without introducing an unacceptably large deviation into data generated by the revised model from the data generated by the original model.